A7A INTERIMREPORT GRANT AFOSR AU)NORHCAROLNA / UNIV AT CHARLO0TE DEPT OF MATHEMATICS M ABDEL-HAMEED AN S DAUG 83 AFOSR-TN 83 AN N

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1 A7A INTERIMREPORT GRANT AFOSR AU)NORHCAROLNA / UNIV AT CHARLO0TE DEPT OF MATHEMATICS M ABDEL-HAMEED AN S DAUG 83 AFOSR-TN 83 AN N AFOSR

2 III III il1.6 MICROCOPY RESOLUTION TEST CHART SC.NATIONAL SUE AU 6 OF STAjOS r9 3 - I 4 IF i *

3 U I UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE (When Date Entered)f REPORT DOCUMENTATrION PAGE READ INSTRUCTIONS urfotfo.e COMPLETING FOR M1. REPORT NUMBER GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG NUMBER SRTR TITLE (and Subtitle) S. TYPE OF REPORT 6 PERIOD COVERED INTERIM: ieport, GRANT AFOSR INTERIM, 1 JUL JUN 83 Q 7. AUTHOR(s) S. CONTRACT OR GRANT NUMBER(&) Mohamed Abdel-Hameed AFOSR PERFORMING O1G. REPORT NUMBER 9 PERFORMING ORGANIZATION N'AME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASK Department of Mathematics & Computer Science AREA A WORK UNIT NUMBERS University of North Carolina PE61102F; 2304/A5 UNCC Station, Charlotte NC C'. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Mathematical & Information Sciences Directorate AUG 83 Air Force Office of Scientific Research Boiling AFB DC NUMBER OF PAGES 14. MONITORING AGENCY NAME & ADDRESS(if different from Controlling Office) IS. SECURITY CLASS. (of this report) UNCLASSIFIED 15a. DECLASSIFICATION' DOWNGRADING SCHEDULE 16. DISTRIBUTION STATEMENT (of this Report) Approved for public release; distribution unlimited. 17. DISTRIBUTION STATEMENT (of the abstract entered it' Block 20, if different fron t Report) 18. SUPPLEMENTARY NOTES 4 EEI. '--.. E 19. KEY WORDS (Continue on reverse aide if necessary and identify by block nuber)a 20. ABSTRACT (Continue on reverse side It necessary and Identify by block number) During the period I Jul Jun 83, the principal investigator and the co- * >.. investigator attended four conferences, presenting papers at two of them. The ",.principal investigator organized a conference on "Stochastic Failure Models, * Replacement and Maintenance Policies, Accelerated Life Testing." He continued - 3 his research on shock models, wear processes, replacement and maintenance policies; revised the paper, "Life Distribution Properties of Devices Subject to a Levy Wear Process" that will appear in Mathematics of Operations Research, wrote the paper, "Pure Jump Damage Processes and submitted it for FORM (CONT.) DD,JAN EDITION OF I NOV65 IS OBSOLETE UNCLASSIFIED 3 1C0 17 U ~d C U RITY CLASSIFICATION OF TNIS PAGE ( 0 ' I ' ala Entered)

4 UNCLASSIFIED SECURITY CLASSIFICATION OF THIS PAGE(Mh Data Entered) I ITEM #LO, CGNT.: publication, and finished the paper, "Pure Jump W.-ar Processes: A Review," which will appear in The Proceedings of the First Saidi Conference on Statistics and its Applications. He also was a key-note speaker at the First Saudi Conference on Statistics. The paper, "Conservative and Dissipative Parts of Non-Measure Preserving Weighted Composition Operators," by the co-investigator, appeared in the Houston Journal of Mathematics. The co-investigator presented a paper on "Approximate Optimal Replacement Policies and Their Stability," at the conference "Stochastic Failure Models, Replacement and Maintenance Policies, Accelerated Life Testing" that was organized by the principal investigator. The details of the research conducted and conferences attended are contained within the interim report. UNCLASSIFIED SECURITY CLASSIFICATION OF T"1 PAGE(Wh Data Entered)

5 AFOSRTh FINAL REPORT GRANT AFOSR From July 1, 1982 to June 30, 1983 / During the period July 1, 1982 to June 30, 1983 the Principal Investigator and the Co-investigator attended four conferences giving papers at two of them. The Principal Investigator organized a Conference on Stochastic Failure Models, Replacement and Maintenance Policies, Accelerated Life Testing". He continued his research on shock models, wear processes, replacement and maintenance policies; revised the paper 'Life Distribution Properties of Devices Subject to a Levy Wear Process that will appear in Mathematics of Operations Research, wrote the paper (Pure Jump Damage Processes and submitted it for publication, and finished the paper "Pure Jump Wear Processes: A Review" which will appear in The Proceedings of the First Saudi Conference on Statistics and -its Applications. He also was a key-note speaker at the First Saudi Conference on Statistics. The paper, 'Conservative and Dissipative Parts of Non-Measure Preserving Weighted Composition Operators". by the Co-investigator, appeared in the Houston Journal of Mathematics. The Co-investigator presented a paper on Approximate Optimal Replacement Policies iand Their Stability, at the conference "Stochastic Failure Models, Replacement and Maintenance Policies, Accelerated Life Testing" that was organized by the Principal Investigator. The details of the research conducted and conferences attended are as follows: Appt-o 41trbut,. w u. t r d II

6 2 I. RESEARCH CONDUCTED A) PURE JUMP DAMAGE PROCESSES. Assume that a device is subject to damage, denoted by X. The damages process is assumed to be an iiicreasing pure jump process. Such a process can be expressed as follows: X t =X 0 +X t where Xd t = Xt (X u - Xu) E Xu -Xuu<t Clearly Xd describes the cumulative wear due to shocks occurring during (O,t] and X 0 is the initial wear. Let (t,k) be the LeVy system for X. It is known from the work of E. Benveniste and J. Jacod that, for each positive Borel measurable function f on R+ x R+ with f(x,x) = 0, for all x in E. E [ Z f(xsxs)] s t =ExJt ds 'R K(X s dy)f(x s y)] Cinlar and Jacod show that there exists a Poisson random measure on R x R whose mean measure element at (s,z) is du dz/z and a deterministic function c such that y: f(xs_,x) = f N(ds,dz)f(Xs_,X s+c(xs_,z)) u~t [0,tJ<R+ almost everywhere for each function f on R+ x R+ of the above type. In particular, it follows that X t = Xo + ] c(x,z)n(ds,dz),.. [O,t]xR+. Chiif. re-,h:,nc,,i InformationDivision The above formula has the following interpretation: t Xt(w) jumps at s if and only if the Poisson random measure N(w,.).-

7 3 has an atom (s,z) and then the jump is from the left hand limit X (w) 5- to the right hand limit Xs = Xs_ + c(x s _,z) The function c(x,z) represents the damage due to a shock of magnitude z occurring at a time when the previous cumulative wear equals x. We determined conditions on the function c and the LeVy kernel K that insure that the life distribution properties of the threshold right tail probability G are inherited as corresponding properties of the survival probability F(t) = E[G (X )]. We note that a stationary LeVy shock and wear t process is a special case of the pure jump processes described above. follows from Ito's representation of every increasing LeVy process as follows: This X = X + f z N(ds,dz) t 0 [O,t]xR+ where N -is a Poisson random measure with mean measure element at (s,dz) equals ds P(dz) and o is a LeVy measure. These results have been submitted for publication. B) OPTIMAL REPLACEMENT AND MAINTENANCE POLICIES FOR DEVICES SUBJECT TO PURE JUMP DAMAGE PROCESSES. Assume that a device is subject to damage occurring randomly in time according to a pure jump process X. Let g(w,t):s2xr+ - R be a function describing the cumulative net yield the device produces till time t. Suppose that g(w,t) = f(x t(w)) for some f:r+ - R. The expected cumulative discounted net yield from observing the device till time t equals Efd(t)f(X )] We give conditions on the functions d and f that ensures the existence of a replacement policy that maximizes the expected cumulative discounted net yield and give a closed form for that replacement time. In the case d(t) - e we show that for a suitable choice of the function f the optimal " a

8 4 replacement policy is a control policy. The following theorem follows from the work of Ikeda and Watanabe or Benveniste and Jacod and the martingale characterization theorem for Poisson random measures due to Jacod. The theorem holds for a larger class of functions than the one indicated. However, the statement given here is sufficient for our purpose. First define Af(x) = f [f(x+c(x,z)) - f(x)] -z R+/{O} z Theorem. Let f:r+ - R be increasing d:r+ - R+ be differentiable. Then, for any stopping time T for which E[ 0 d'(t)f(xt )dt ] exists and is finite we have that E[d(T)f(XT)] - d(o)f(x) =E[ ft d'(t)f(x )dtj + E[ ft d(t)af(x )dtl We proved the following: Theorem. Let f be concave increasing and c(x,z) be decreasing in the first argument. Let d be decreasing and concave. Let F be the class of stopping times for which E[ ft d'(t)f(xt)dt] exists and is finite. Define u(t) = d'(t'f(x t) + d(t)af(x t) Let =* infit:u(t) < 0} Then T* is optimal in the sense that E[d(T*)f(XT*)I = sup E[d(T)f(XT) TLF Theorem. Let f be concave increasing and c(x,z) be decreasing in the first argument. Let d(s) = e, > 0. Let F 1 be the class of ' V.

9 5 stopping times for which E[ ft e- Atf(X )dt] exists and is finite. Define 0 t A = {x:af(x) - Af(x) < 01. Then the stopping time T* = inf{t:xt ca is optimal in the sense that E[e- T*f(XT*)] = sup E[e- Tf(XT) ] TEF 1 C) A PRACTICAL DEFINITION OF STABILITY FOR OPTIMAL STOPPING TIMES. The results of this work represent progress on question 1(C) of the proposal. This question asked whether the two main types of stability are equivalent. While we do not provide a complete answer, the result which is obtained indicates that in a practical sense they can be considered to be equivalent. Briefly, a problem (X,g) is considered to be center stable, here X is a process and g is a reward function defined on the state space of X, if a close to optimal solution to the problem (X,g) is close to optimal for any problem (X9 1,g) which is sufficiently "close" to the problem (X,g). The other broad class of stability considered views (X,g) as "stable" if a close to optimal solution to the problem (X,gl) is close to optimal for (X,g) provided (X9 1,g) is "close enough" to (X,g) We say that the problem (X,g) is weakly-payoff stable at x if for each L > 0 there exists a positive integer n 1 such that if X is "close enough" 0 to X then sup E )g(x,) > 7 g(x) - L (here 71g is the smallest excessive majorant of g with respect to X.) The problem (X,g) is said to be stable with respect to bounded times if for every c > 0 and n 0 {1,2,... there O x1 is a I, > 0 and an a > 0 such that every a-optimal time T < n for (X,g 1 ) is c-optimal for (X,g) whenever (X,g 1 ) is within a of (X,g) The,,I I -,! i i l l~ i,.: I......

10 6 concept of center stable with respect to bounded times is defined analagously. The main result is the following: THEOREM 1. The following are equivalent: a) (X,g) is center stable. b) (X,g) is center stable with respect to bounded times. c) (X,g) is weakly-payoff stable. d) (X,g) is weakly-payoff stable and stable with respect to bounded times. This research intended to incorporate this result into a rewrite of his paper entitled, "Stability of Optimal Stopping Problems for Markov Processes". D) STABILITY OF OPTIMAL REPLACEMENT PROBLEMS. The results in this work represent progress on one of the main problems of the proposal, namely that of developing a stability theory for optimal replacement problems for wear processes which allows the modeling processes to have -different lifetimes. -* One considers a nonterminating wear process X, a fixed stopping time F * with uniformly bounded expected lifetime, a family N() of stopping times which are bounded above by r and a family SX of related nonterminating processes for all of which F has uniformly bounded expected lifetime. A metric is placed on SX x N(F) which reflects closeness of the processes in S X and of the stopping times in N(F) A problem (X,C,g) is considered to be stable if a close to optimal replacement solution to the problem (X,,g) is close to optimal for any sufficient close problem (X',(,gl) and vice-versa. If the lifetimes are fixed, we have stability. However, examples show that stability of replacement problems is quite sensitive to variation in the i"stability lifetimes. These results are currently being written up in a paper entitled, of Optimal Replacement Problems". II*,

11 7 E) AN ITERATION SCHEME FOR OBTAINING APPROXIMATE OPTIMAL REPLACEMENT POLICIES. In this work we analyze the following iterative technique. Let bl = 1I E g(x ) E where (X,c,g) is a wear process with lifetime C. Consider the E 0 () problem of maximizing the criterion (bl,r) = bie 0 (T) - Eog(XT) for c < If we are interested in an c-optimal policy, then tolerances a > 0 and > 0 are determined in terms of X,g,r and c so that, if a -optimal policy 1l for the problem i(b 1,,) gives x(bl, I) < a then is already E g(x l) c-optimal, otherwise take b 2 = E ( ) and repeat the steps on the criterion 2 E0(Ti $(b 2 9 = b 2 E (0) - Eog(Xr) Under very reasonable assumptions on g (it must be positive and bounded away from zero), it is shown that this iterative method will always supply an t:-optimal replacement policy. This method has several advantages over those being employed in the literature where enough is assumed about the process to associate a contraction mapping with the optimal replacement problem and/or the function g is so specialized as to permit the identification of a control policy by inspection of an associated formula employing the generator of the semigroup associated with X. This iterative method needs no such assumptions on g and works even when a true o- optimal policy does not exist. The lack of a o-optimal policy would frustrate all cases in the literature so far; especially the contraction mapping approach. These results are written up in the paper entitled, "An Iterative Scheme for Approximating Optimal Repl,ement Pnlicies". Il. MEETINGS ATTENDED 1. The First Saudi Conference on Statistics and Applications, May I - May 5, Denver meeting of the A.M.S. E '

12 8 3. April meeting of Seminar on Probability in Gainesville, Florida. 4. June meeting on Stochastic Failure Models in Charlotte, North Carolina (Gave a talk entitled, "Approximate Optimal Replacement Policies and Their Stability".) i.

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